LMFDB - Embedded newform 3808.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3808,2,Mod(1,3808)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3808, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3808.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$30.4070330897$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.147697840.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 10x^{3} + 18x^{2} - 16x + 3 $ x^6 - 2x^5 - 10x^4 + 10x^3 + 18x^2 - 16*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.18379$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.18379 q^{3} -3.11205 q^{5} +1.00000 q^{7} +7.13652 q^{9} +1.84951 q^{11} -5.61227 q^{13} +9.90811 q^{15} +1.00000 q^{17} -4.50640 q^{19} -3.18379 q^{21} -6.03193 q^{23} +4.68484 q^{25} -13.1698 q^{27} +7.37503 q^{29} +0.507231 q^{31} -5.88844 q^{33} -3.11205 q^{35} +10.3544 q^{37} +17.8683 q^{39} +0.176337 q^{41} -2.57897 q^{43} -22.2092 q^{45} +9.01133 q^{47} +1.00000 q^{49} -3.18379 q^{51} +9.46796 q^{53} -5.75576 q^{55} +14.3474 q^{57} -3.36714 q^{59} +9.30246 q^{61} +7.13652 q^{63} +17.4657 q^{65} -3.89541 q^{67} +19.2044 q^{69} -9.87609 q^{71} -1.50777 q^{73} -14.9156 q^{75} +1.84951 q^{77} +11.8534 q^{79} +20.5204 q^{81} -1.70681 q^{83} -3.11205 q^{85} -23.4806 q^{87} -7.48581 q^{89} -5.61227 q^{91} -1.61492 q^{93} +14.0241 q^{95} +13.9046 q^{97} +13.1990 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{21} - 6 q^{23} + 12 q^{25} - 10 q^{27} + 4 q^{29} + 8 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} + 18 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^13 + 6 * q^17 - 18 * q^19 - 4 * q^21 - 6 * q^23 + 12 * q^25 - 10 * q^27 + 4 * q^29 + 8 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 + 18 * q^39 - 4 * q^41 - 20 * q^43 - 24 * q^45 - 6 * q^47 + 6 * q^49 - 4 * q^51 + 14 * q^53 - 4 * q^55 - 4 * q^57 - 20 * q^59 + 2 * q^61 + 8 * q^63 + 6 * q^65 - 28 * q^67 - 4 * q^69 - 18 * q^71 + 4 * q^73 - 20 * q^75 - 8 * q^77 + 2 * q^79 + 34 * q^81 - 20 * q^83 - 4 * q^85 - 22 * q^87 - 6 * q^89 - 4 * q^91 + 40 * q^93 + 6 * q^95 + 30 * q^97 - 32 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.18379	−1.83816	−0.919081	−	0.394068i	$-0.871067\pi$
$3$	−3.18379	−1.83816	−0.919081	+	0.394068i	$0.871067\pi$
$4$	0	0
$5$	−3.11205	−1.39175	−0.695875	−	0.718163i	$-0.744983\pi$
$5$	−3.11205	−1.39175	−0.695875	+	0.718163i	$0.744983\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	7.13652	2.37884
$10$	0	0
$11$	1.84951	0.557647	0.278824	−	0.960342i	$-0.410055\pi$
$11$	1.84951	0.557647	0.278824	+	0.960342i	$0.410055\pi$
$12$	0	0
$13$	−5.61227	−1.55656	−0.778282	−	0.627915i	$-0.783909\pi$
$13$	−5.61227	−1.55656	−0.778282	+	0.627915i	$0.783909\pi$
$14$	0	0
$15$	9.90811	2.55826
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−4.50640	−1.03384	−0.516920	−	0.856034i	$-0.672921\pi$
$19$	−4.50640	−1.03384	−0.516920	+	0.856034i	$0.672921\pi$
$20$	0	0
$21$	−3.18379	−0.694760
$22$	0	0
$23$	−6.03193	−1.25774	−0.628872	−	0.777509i	$-0.716483\pi$
$23$	−6.03193	−1.25774	−0.628872	+	0.777509i	$0.716483\pi$
$24$	0	0
$25$	4.68484	0.936969
$26$	0	0
$27$	−13.1698	−2.53453
$28$	0	0
$29$	7.37503	1.36951	0.684755	−	0.728774i	$-0.259909\pi$
$29$	7.37503	1.36951	0.684755	+	0.728774i	$0.259909\pi$
$30$	0	0
$31$	0.507231	0.0911014	0.0455507	−	0.998962i	$-0.485496\pi$
$31$	0.507231	0.0911014	0.0455507	+	0.998962i	$0.485496\pi$
$32$	0	0
$33$	−5.88844	−1.02505
$34$	0	0
$35$	−3.11205	−0.526032
$36$	0	0
$37$	10.3544	1.70226	0.851130	−	0.524955i	$-0.175918\pi$
$37$	10.3544	1.70226	0.851130	+	0.524955i	$0.175918\pi$
$38$	0	0
$39$	17.8683	2.86122
$40$	0	0
$41$	0.176337	0.0275392	0.0137696	−	0.999905i	$-0.495617\pi$
$41$	0.176337	0.0275392	0.0137696	+	0.999905i	$0.495617\pi$
$42$	0	0
$43$	−2.57897	−0.393290	−0.196645	−	0.980475i	$-0.563005\pi$
$43$	−2.57897	−0.393290	−0.196645	+	0.980475i	$0.563005\pi$
$44$	0	0
$45$	−22.2092	−3.31075
$46$	0	0
$47$	9.01133	1.31444	0.657219	−	0.753700i	$-0.271733\pi$
$47$	9.01133	1.31444	0.657219	+	0.753700i	$0.271733\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.18379	−0.445820
$52$	0	0
$53$	9.46796	1.30052	0.650262	−	0.759710i	$-0.274659\pi$
$53$	9.46796	1.30052	0.650262	+	0.759710i	$0.274659\pi$
$54$	0	0
$55$	−5.75576	−0.776106
$56$	0	0
$57$	14.3474	1.90036
$58$	0	0
$59$	−3.36714	−0.438364	−0.219182	−	0.975684i	$-0.570339\pi$
$59$	−3.36714	−0.438364	−0.219182	+	0.975684i	$0.570339\pi$
$60$	0	0
$61$	9.30246	1.19106	0.595529	−	0.803334i	$-0.296942\pi$
$61$	9.30246	1.19106	0.595529	+	0.803334i	$0.296942\pi$
$62$	0	0
$63$	7.13652	0.899117
$64$	0	0
$65$	17.4657	2.16635
$66$	0	0
$67$	−3.89541	−0.475900	−0.237950	−	0.971277i	$-0.576475\pi$
$67$	−3.89541	−0.475900	−0.237950	+	0.971277i	$0.576475\pi$
$68$	0	0
$69$	19.2044	2.31194
$70$	0	0
$71$	−9.87609	−1.17208	−0.586038	−	0.810284i	$-0.699313\pi$
$71$	−9.87609	−1.17208	−0.586038	+	0.810284i	$0.699313\pi$
$72$	0	0
$73$	−1.50777	−0.176471	−0.0882356	−	0.996100i	$-0.528123\pi$
$73$	−1.50777	−0.176471	−0.0882356	+	0.996100i	$0.528123\pi$
$74$	0	0
$75$	−14.9156	−1.72230
$76$	0	0
$77$	1.84951	0.210771
$78$	0	0
$79$	11.8534	1.33361	0.666805	−	0.745232i	$-0.267661\pi$
$79$	11.8534	1.33361	0.666805	+	0.745232i	$0.267661\pi$
$80$	0	0
$81$	20.5204	2.28004
$82$	0	0
$83$	−1.70681	−0.187347	−0.0936733	−	0.995603i	$-0.529861\pi$
$83$	−1.70681	−0.187347	−0.0936733	+	0.995603i	$0.529861\pi$
$84$	0	0
$85$	−3.11205	−0.337549
$86$	0	0
$87$	−23.4806	−2.51738
$88$	0	0
$89$	−7.48581	−0.793494	−0.396747	−	0.917928i	$-0.629861\pi$
$89$	−7.48581	−0.793494	−0.396747	+	0.917928i	$0.629861\pi$
$90$	0	0
$91$	−5.61227	−0.588326
$92$	0	0
$93$	−1.61492	−0.167459
$94$	0	0
$95$	14.0241	1.43885
$96$	0	0
$97$	13.9046	1.41180	0.705901	−	0.708311i	$-0.250542\pi$
$97$	13.9046	1.41180	0.705901	+	0.708311i	$0.250542\pi$
$98$	0	0
$99$	13.1990	1.32655
$100$	0	0
$101$	12.4295	1.23678	0.618391	−	0.785870i	$-0.287785\pi$
$101$	12.4295	1.23678	0.618391	+	0.785870i	$0.287785\pi$
$102$	0	0
$103$	15.3495	1.51244	0.756218	−	0.654320i	$-0.227045\pi$
$103$	15.3495	1.51244	0.756218	+	0.654320i	$0.227045\pi$
$104$	0	0
$105$	9.90811	0.966933
$106$	0	0
$107$	−14.4365	−1.39563	−0.697815	−	0.716278i	$-0.745844\pi$
$107$	−14.4365	−1.39563	−0.697815	+	0.716278i	$0.745844\pi$
$108$	0	0
$109$	3.25636	0.311903	0.155952	−	0.987765i	$-0.450156\pi$
$109$	3.25636	0.311903	0.155952	+	0.987765i	$0.450156\pi$
$110$	0	0
$111$	−32.9664	−3.12903
$112$	0	0
$113$	−12.3428	−1.16111	−0.580555	−	0.814221i	$-0.697164\pi$
$113$	−12.3428	−1.16111	−0.580555	+	0.814221i	$0.697164\pi$
$114$	0	0
$115$	18.7717	1.75047
$116$	0	0
$117$	−40.0521	−3.70282
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−7.57932	−0.689029
$122$	0	0
$123$	−0.561419	−0.0506214
$124$	0	0
$125$	0.980779	0.0877235
$126$	0	0
$127$	−13.3797	−1.18726	−0.593630	−	0.804738i	$-0.702306\pi$
$127$	−13.3797	−1.18726	−0.593630	+	0.804738i	$0.702306\pi$
$128$	0	0
$129$	8.21091	0.722930
$130$	0	0
$131$	21.3868	1.86857	0.934287	−	0.356521i	$-0.116037\pi$
$131$	21.3868	1.86857	0.934287	+	0.356521i	$0.116037\pi$
$132$	0	0
$133$	−4.50640	−0.390754
$134$	0	0
$135$	40.9851	3.52744
$136$	0	0
$137$	−18.2721	−1.56109	−0.780546	−	0.625098i	$-0.785059\pi$
$137$	−18.2721	−1.56109	−0.780546	+	0.625098i	$0.785059\pi$
$138$	0	0
$139$	6.82534	0.578918	0.289459	−	0.957190i	$-0.406525\pi$
$139$	6.82534	0.578918	0.289459	+	0.957190i	$0.406525\pi$
$140$	0	0
$141$	−28.6902	−2.41615
$142$	0	0
$143$	−10.3799	−0.868014
$144$	0	0
$145$	−22.9515	−1.90602
$146$	0	0
$147$	−3.18379	−0.262595
$148$	0	0
$149$	−1.31560	−0.107778	−0.0538891	−	0.998547i	$-0.517162\pi$
$149$	−1.31560	−0.107778	−0.0538891	+	0.998547i	$0.517162\pi$
$150$	0	0
$151$	−14.6945	−1.19582	−0.597911	−	0.801562i	$-0.704003\pi$
$151$	−14.6945	−1.19582	−0.597911	+	0.801562i	$0.704003\pi$
$152$	0	0
$153$	7.13652	0.576954
$154$	0	0
$155$	−1.57853	−0.126790
$156$	0	0
$157$	−24.7087	−1.97197	−0.985984	−	0.166840i	$-0.946644\pi$
$157$	−24.7087	−1.97197	−0.985984	+	0.166840i	$0.946644\pi$
$158$	0	0
$159$	−30.1440	−2.39057
$160$	0	0
$161$	−6.03193	−0.475382
$162$	0	0
$163$	−8.34355	−0.653517	−0.326759	−	0.945108i	$-0.605956\pi$
$163$	−8.34355	−0.653517	−0.326759	+	0.945108i	$0.605956\pi$
$164$	0	0
$165$	18.3251	1.42661
$166$	0	0
$167$	−17.6987	−1.36957	−0.684785	−	0.728745i	$-0.740104\pi$
$167$	−17.6987	−1.36957	−0.684785	+	0.728745i	$0.740104\pi$
$168$	0	0
$169$	18.4976	1.42289
$170$	0	0
$171$	−32.1600	−2.45934
$172$	0	0
$173$	8.68450	0.660271	0.330135	−	0.943934i	$-0.392906\pi$
$173$	8.68450	0.660271	0.330135	+	0.943934i	$0.392906\pi$
$174$	0	0
$175$	4.68484	0.354141
$176$	0	0
$177$	10.7203	0.805784
$178$	0	0
$179$	−13.2652	−0.991487	−0.495743	−	0.868469i	$-0.665104\pi$
$179$	−13.2652	−0.991487	−0.495743	+	0.868469i	$0.665104\pi$
$180$	0	0
$181$	4.48091	0.333063	0.166532	−	0.986036i	$-0.446743\pi$
$181$	4.48091	0.333063	0.166532	+	0.986036i	$0.446743\pi$
$182$	0	0
$183$	−29.6171	−2.18936
$184$	0	0
$185$	−32.2235	−2.36912
$186$	0	0
$187$	1.84951	0.135249
$188$	0	0
$189$	−13.1698	−0.957963
$190$	0	0
$191$	4.47473	0.323780	0.161890	−	0.986809i	$-0.448241\pi$
$191$	4.47473	0.323780	0.161890	+	0.986809i	$0.448241\pi$
$192$	0	0
$193$	−7.97711	−0.574205	−0.287103	−	0.957900i	$-0.592692\pi$
$193$	−7.97711	−0.574205	−0.287103	+	0.957900i	$0.592692\pi$
$194$	0	0
$195$	−55.6070	−3.98210
$196$	0	0
$197$	−12.2507	−0.872825	−0.436412	−	0.899747i	$-0.643751\pi$
$197$	−12.2507	−0.872825	−0.436412	+	0.899747i	$0.643751\pi$
$198$	0	0
$199$	8.82009	0.625240	0.312620	−	0.949878i	$-0.398793\pi$
$199$	8.82009	0.625240	0.312620	+	0.949878i	$0.398793\pi$
$200$	0	0
$201$	12.4022	0.874781
$202$	0	0
$203$	7.37503	0.517626
$204$	0	0
$205$	−0.548768	−0.0383276
$206$	0	0
$207$	−43.0470	−2.99197
$208$	0	0
$209$	−8.33462	−0.576518
$210$	0	0
$211$	3.37102	0.232070	0.116035	−	0.993245i	$-0.462981\pi$
$211$	3.37102	0.232070	0.116035	+	0.993245i	$0.462981\pi$
$212$	0	0
$213$	31.4434	2.15447
$214$	0	0
$215$	8.02589	0.547361
$216$	0	0
$217$	0.507231	0.0344331
$218$	0	0
$219$	4.80042	0.324383
$220$	0	0
$221$	−5.61227	−0.377522
$222$	0	0
$223$	19.5883	1.31173	0.655866	−	0.754877i	$-0.272303\pi$
$223$	19.5883	1.31173	0.655866	+	0.754877i	$0.272303\pi$
$224$	0	0
$225$	33.4335	2.22890
$226$	0	0
$227$	6.89634	0.457726	0.228863	−	0.973459i	$-0.426499\pi$
$227$	6.89634	0.457726	0.228863	+	0.973459i	$0.426499\pi$
$228$	0	0
$229$	13.1425	0.868478	0.434239	−	0.900798i	$-0.357017\pi$
$229$	13.1425	0.868478	0.434239	+	0.900798i	$0.357017\pi$
$230$	0	0
$231$	−5.88844	−0.387431
$232$	0	0
$233$	18.9448	1.24111	0.620557	−	0.784161i	$-0.286906\pi$
$233$	18.9448	1.24111	0.620557	+	0.784161i	$0.286906\pi$
$234$	0	0
$235$	−28.0437	−1.82937
$236$	0	0
$237$	−37.7387	−2.45139
$238$	0	0
$239$	7.26079	0.469661	0.234831	−	0.972036i	$-0.424546\pi$
$239$	7.26079	0.469661	0.234831	+	0.972036i	$0.424546\pi$
$240$	0	0
$241$	−6.38077	−0.411021	−0.205511	−	0.978655i	$-0.565886\pi$
$241$	−6.38077	−0.411021	−0.205511	+	0.978655i	$0.565886\pi$
$242$	0	0
$243$	−25.8231	−1.65655
$244$	0	0
$245$	−3.11205	−0.198821
$246$	0	0
$247$	25.2911	1.60924
$248$	0	0
$249$	5.43412	0.344373
$250$	0	0
$251$	−26.1642	−1.65147	−0.825736	−	0.564057i	$-0.809240\pi$
$251$	−26.1642	−1.65147	−0.825736	+	0.564057i	$0.809240\pi$
$252$	0	0
$253$	−11.1561	−0.701378
$254$	0	0
$255$	9.90811	0.620470
$256$	0	0
$257$	10.1670	0.634198	0.317099	−	0.948392i	$-0.397291\pi$
$257$	10.1670	0.634198	0.317099	+	0.948392i	$0.397291\pi$
$258$	0	0
$259$	10.3544	0.643394
$260$	0	0
$261$	52.6321	3.25785
$262$	0	0
$263$	−24.4026	−1.50473	−0.752365	−	0.658747i	$-0.771087\pi$
$263$	−24.4026	−1.50473	−0.752365	+	0.658747i	$0.771087\pi$
$264$	0	0
$265$	−29.4647	−1.81000
$266$	0	0
$267$	23.8332	1.45857
$268$	0	0
$269$	−11.4364	−0.697292	−0.348646	−	0.937254i	$-0.613358\pi$
$269$	−11.4364	−0.697292	−0.348646	+	0.937254i	$0.613358\pi$
$270$	0	0
$271$	4.11951	0.250242	0.125121	−	0.992141i	$-0.460068\pi$
$271$	4.11951	0.250242	0.125121	+	0.992141i	$0.460068\pi$
$272$	0	0
$273$	17.8683	1.08144
$274$	0	0
$275$	8.66465	0.522498
$276$	0	0
$277$	−17.5133	−1.05227	−0.526135	−	0.850401i	$-0.676360\pi$
$277$	−17.5133	−1.05227	−0.526135	+	0.850401i	$0.676360\pi$
$278$	0	0
$279$	3.61987	0.216716
$280$	0	0
$281$	−20.3210	−1.21225	−0.606125	−	0.795370i	$-0.707277\pi$
$281$	−20.3210	−1.21225	−0.606125	+	0.795370i	$0.707277\pi$
$282$	0	0
$283$	−2.90331	−0.172584	−0.0862919	−	0.996270i	$-0.527502\pi$
$283$	−2.90331	−0.172584	−0.0862919	+	0.996270i	$0.527502\pi$
$284$	0	0
$285$	−44.6499	−2.64483
$286$	0	0
$287$	0.176337	0.0104088
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−44.2694	−2.59512
$292$	0	0
$293$	−22.4801	−1.31330	−0.656651	−	0.754194i	$-0.728028\pi$
$293$	−22.4801	−1.31330	−0.656651	+	0.754194i	$0.728028\pi$
$294$	0	0
$295$	10.4787	0.610093
$296$	0	0
$297$	−24.3577	−1.41338
$298$	0	0
$299$	33.8528	1.95776
$300$	0	0
$301$	−2.57897	−0.148650
$302$	0	0
$303$	−39.5730	−2.27341
$304$	0	0
$305$	−28.9497	−1.65766
$306$	0	0
$307$	4.06046	0.231743	0.115871	−	0.993264i	$-0.463034\pi$
$307$	4.06046	0.231743	0.115871	+	0.993264i	$0.463034\pi$
$308$	0	0
$309$	−48.8697	−2.78010
$310$	0	0
$311$	−25.0053	−1.41792	−0.708959	−	0.705250i	$-0.750835\pi$
$311$	−25.0053	−1.41792	−0.708959	+	0.705250i	$0.750835\pi$
$312$	0	0
$313$	−18.7763	−1.06130	−0.530650	−	0.847591i	$-0.678052\pi$
$313$	−18.7763	−1.06130	−0.530650	+	0.847591i	$0.678052\pi$
$314$	0	0
$315$	−22.2092	−1.25135
$316$	0	0
$317$	23.6062	1.32586	0.662929	−	0.748682i	$-0.269313\pi$
$317$	23.6062	1.32586	0.662929	+	0.748682i	$0.269313\pi$
$318$	0	0
$319$	13.6402	0.763703
$320$	0	0
$321$	45.9629	2.56540
$322$	0	0
$323$	−4.50640	−0.250743
$324$	0	0
$325$	−26.2926	−1.45845
$326$	0	0
$327$	−10.3676	−0.573329
$328$	0	0
$329$	9.01133	0.496811
$330$	0	0
$331$	−9.00795	−0.495122	−0.247561	−	0.968872i	$-0.579629\pi$
$331$	−9.00795	−0.495122	−0.247561	+	0.968872i	$0.579629\pi$
$332$	0	0
$333$	73.8947	4.04940
$334$	0	0
$335$	12.1227	0.662333
$336$	0	0
$337$	12.8813	0.701689	0.350844	−	0.936434i	$-0.385895\pi$
$337$	12.8813	0.701689	0.350844	+	0.936434i	$0.385895\pi$
$338$	0	0
$339$	39.2968	2.13431
$340$	0	0
$341$	0.938128	0.0508025
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−59.7650	−3.21764
$346$	0	0
$347$	4.92381	0.264324	0.132162	−	0.991228i	$-0.457808\pi$
$347$	4.92381	0.264324	0.132162	+	0.991228i	$0.457808\pi$
$348$	0	0
$349$	−12.8085	−0.685622	−0.342811	−	0.939404i	$-0.611379\pi$
$349$	−12.8085	−0.685622	−0.342811	+	0.939404i	$0.611379\pi$
$350$	0	0
$351$	73.9126	3.94516
$352$	0	0
$353$	5.80004	0.308705	0.154352	−	0.988016i	$-0.450671\pi$
$353$	5.80004	0.308705	0.154352	+	0.988016i	$0.450671\pi$
$354$	0	0
$355$	30.7349	1.63124
$356$	0	0
$357$	−3.18379	−0.168504
$358$	0	0
$359$	21.6651	1.14344	0.571719	−	0.820450i	$-0.306277\pi$
$359$	21.6651	1.14344	0.571719	+	0.820450i	$0.306277\pi$
$360$	0	0
$361$	1.30764	0.0688231
$362$	0	0
$363$	24.1310	1.26655
$364$	0	0
$365$	4.69225	0.245604
$366$	0	0
$367$	−4.80370	−0.250751	−0.125375	−	0.992109i	$-0.540014\pi$
$367$	−4.80370	−0.250751	−0.125375	+	0.992109i	$0.540014\pi$
$368$	0	0
$369$	1.25843	0.0655113
$370$	0	0
$371$	9.46796	0.491552
$372$	0	0
$373$	−8.56349	−0.443401	−0.221700	−	0.975115i	$-0.571161\pi$
$373$	−8.56349	−0.443401	−0.221700	+	0.975115i	$0.571161\pi$
$374$	0	0
$375$	−3.12259	−0.161250
$376$	0	0
$377$	−41.3907	−2.13173
$378$	0	0
$379$	21.9474	1.12736	0.563681	−	0.825992i	$-0.309385\pi$
$379$	21.9474	1.12736	0.563681	+	0.825992i	$0.309385\pi$
$380$	0	0
$381$	42.5983	2.18238
$382$	0	0
$383$	−37.0174	−1.89150	−0.945751	−	0.324892i	$-0.894672\pi$
$383$	−37.0174	−1.89150	−0.945751	+	0.324892i	$0.894672\pi$
$384$	0	0
$385$	−5.75576	−0.293340
$386$	0	0
$387$	−18.4049	−0.935574
$388$	0	0
$389$	23.2409	1.17836	0.589181	−	0.808001i	$-0.299451\pi$
$389$	23.2409	1.17836	0.589181	+	0.808001i	$0.299451\pi$
$390$	0	0
$391$	−6.03193	−0.305048
$392$	0	0
$393$	−68.0911	−3.43474
$394$	0	0
$395$	−36.8883	−1.85605
$396$	0	0
$397$	−5.62311	−0.282216	−0.141108	−	0.989994i	$-0.545066\pi$
$397$	−5.62311	−0.282216	−0.141108	+	0.989994i	$0.545066\pi$
$398$	0	0
$399$	14.3474	0.718270
$400$	0	0
$401$	20.3066	1.01406	0.507032	−	0.861927i	$-0.330742\pi$
$401$	20.3066	1.01406	0.507032	+	0.861927i	$0.330742\pi$
$402$	0	0
$403$	−2.84672	−0.141805
$404$	0	0
$405$	−63.8604	−3.17325
$406$	0	0
$407$	19.1506	0.949261
$408$	0	0
$409$	14.5104	0.717492	0.358746	−	0.933435i	$-0.383204\pi$
$409$	14.5104	0.717492	0.358746	+	0.933435i	$0.383204\pi$
$410$	0	0
$411$	58.1746	2.86954
$412$	0	0
$413$	−3.36714	−0.165686
$414$	0	0
$415$	5.31167	0.260740
$416$	0	0
$417$	−21.7305	−1.06415
$418$	0	0
$419$	−32.3875	−1.58223	−0.791116	−	0.611667i	$-0.790499\pi$
$419$	−32.3875	−1.58223	−0.791116	+	0.611667i	$0.790499\pi$
$420$	0	0
$421$	5.30325	0.258464	0.129232	−	0.991614i	$-0.458749\pi$
$421$	5.30325	0.258464	0.129232	+	0.991614i	$0.458749\pi$
$422$	0	0
$423$	64.3096	3.12684
$424$	0	0
$425$	4.68484	0.227248
$426$	0	0
$427$	9.30246	0.450178
$428$	0	0
$429$	33.0475	1.59555
$430$	0	0
$431$	24.3623	1.17349	0.586746	−	0.809771i	$-0.300409\pi$
$431$	24.3623	1.17349	0.586746	+	0.809771i	$0.300409\pi$
$432$	0	0
$433$	19.0987	0.917826	0.458913	−	0.888481i	$-0.348239\pi$
$433$	19.0987	0.917826	0.458913	+	0.888481i	$0.348239\pi$
$434$	0	0
$435$	73.0727	3.50357
$436$	0	0
$437$	27.1823	1.30030
$438$	0	0
$439$	34.9043	1.66589	0.832946	−	0.553354i	$-0.186652\pi$
$439$	34.9043	1.66589	0.832946	+	0.553354i	$0.186652\pi$
$440$	0	0
$441$	7.13652	0.339834
$442$	0	0
$443$	−39.1983	−1.86237	−0.931183	−	0.364552i	$-0.881222\pi$
$443$	−39.1983	−1.86237	−0.931183	+	0.364552i	$0.881222\pi$
$444$	0	0
$445$	23.2962	1.10435
$446$	0	0
$447$	4.18860	0.198114
$448$	0	0
$449$	−18.4030	−0.868493	−0.434246	−	0.900794i	$-0.642985\pi$
$449$	−18.4030	−0.868493	−0.434246	+	0.900794i	$0.642985\pi$
$450$	0	0
$451$	0.326136	0.0153571
$452$	0	0
$453$	46.7843	2.19812
$454$	0	0
$455$	17.4657	0.818803
$456$	0	0
$457$	−11.7635	−0.550271	−0.275136	−	0.961405i	$-0.588723\pi$
$457$	−11.7635	−0.550271	−0.275136	+	0.961405i	$0.588723\pi$
$458$	0	0
$459$	−13.1698	−0.614715
$460$	0	0
$461$	−33.2983	−1.55086	−0.775428	−	0.631436i	$-0.782466\pi$
$461$	−33.2983	−1.55086	−0.775428	+	0.631436i	$0.782466\pi$
$462$	0	0
$463$	−5.68338	−0.264129	−0.132065	−	0.991241i	$-0.542161\pi$
$463$	−5.68338	−0.264129	−0.132065	+	0.991241i	$0.542161\pi$
$464$	0	0
$465$	5.02570	0.233061
$466$	0	0
$467$	−38.3971	−1.77681	−0.888403	−	0.459064i	$-0.848185\pi$
$467$	−38.3971	−1.77681	−0.888403	+	0.459064i	$0.848185\pi$
$468$	0	0
$469$	−3.89541	−0.179873
$470$	0	0
$471$	78.6673	3.62480
$472$	0	0
$473$	−4.76983	−0.219317
$474$	0	0
$475$	−21.1118	−0.968675
$476$	0	0
$477$	67.5683	3.09374
$478$	0	0
$479$	−3.56259	−0.162779	−0.0813893	−	0.996682i	$-0.525936\pi$
$479$	−3.56259	−0.162779	−0.0813893	+	0.996682i	$0.525936\pi$
$480$	0	0
$481$	−58.1119	−2.64968
$482$	0	0
$483$	19.2044	0.873830
$484$	0	0
$485$	−43.2719	−1.96488
$486$	0	0
$487$	23.5851	1.06874	0.534371	−	0.845250i	$-0.320548\pi$
$487$	23.5851	1.06874	0.534371	+	0.845250i	$0.320548\pi$
$488$	0	0
$489$	26.5641	1.20127
$490$	0	0
$491$	−14.1037	−0.636492	−0.318246	−	0.948008i	$-0.603094\pi$
$491$	−14.1037	−0.636492	−0.318246	+	0.948008i	$0.603094\pi$
$492$	0	0
$493$	7.37503	0.332155
$494$	0	0
$495$	−41.0761	−1.84623
$496$	0	0
$497$	−9.87609	−0.443003
$498$	0	0
$499$	32.3990	1.45038	0.725188	−	0.688551i	$-0.241753\pi$
$499$	32.3990	1.45038	0.725188	+	0.688551i	$0.241753\pi$
$500$	0	0
$501$	56.3491	2.51749
$502$	0	0
$503$	−16.5150	−0.736367	−0.368184	−	0.929753i	$-0.620020\pi$
$503$	−16.5150	−0.736367	−0.368184	+	0.929753i	$0.620020\pi$
$504$	0	0
$505$	−38.6812	−1.72129
$506$	0	0
$507$	−58.8924	−2.61551
$508$	0	0
$509$	30.9498	1.37183	0.685913	−	0.727684i	$-0.259403\pi$
$509$	30.9498	1.37183	0.685913	+	0.727684i	$0.259403\pi$
$510$	0	0
$511$	−1.50777	−0.0666998
$512$	0	0
$513$	59.3485	2.62030
$514$	0	0
$515$	−47.7685	−2.10493
$516$	0	0
$517$	16.6665	0.732993
$518$	0	0
$519$	−27.6496	−1.21368
$520$	0	0
$521$	−33.4868	−1.46708	−0.733541	−	0.679645i	$-0.762134\pi$
$521$	−33.4868	−1.46708	−0.733541	+	0.679645i	$0.762134\pi$
$522$	0	0
$523$	10.9080	0.476974	0.238487	−	0.971146i	$-0.423348\pi$
$523$	10.9080	0.476974	0.238487	+	0.971146i	$0.423348\pi$
$524$	0	0
$525$	−14.9156	−0.650969
$526$	0	0
$527$	0.507231	0.0220953
$528$	0	0
$529$	13.3842	0.581920
$530$	0	0
$531$	−24.0296	−1.04280
$532$	0	0
$533$	−0.989649	−0.0428664
$534$	0	0
$535$	44.9271	1.94237
$536$	0	0
$537$	42.2336	1.82251
$538$	0	0
$539$	1.84951	0.0796639
$540$	0	0
$541$	−39.9806	−1.71890	−0.859450	−	0.511219i	$-0.829194\pi$
$541$	−39.9806	−1.71890	−0.859450	+	0.511219i	$0.829194\pi$
$542$	0	0
$543$	−14.2663	−0.612224
$544$	0	0
$545$	−10.1340	−0.434091
$546$	0	0
$547$	−14.2550	−0.609502	−0.304751	−	0.952432i	$-0.598573\pi$
$547$	−14.2550	−0.609502	−0.304751	+	0.952432i	$0.598573\pi$
$548$	0	0
$549$	66.3872	2.83334
$550$	0	0
$551$	−33.2349	−1.41585
$552$	0	0
$553$	11.8534	0.504057
$554$	0	0
$555$	102.593	4.35483
$556$	0	0
$557$	−31.5818	−1.33817	−0.669083	−	0.743188i	$-0.733313\pi$
$557$	−31.5818	−1.33817	−0.669083	+	0.743188i	$0.733313\pi$
$558$	0	0
$559$	14.4739	0.612181
$560$	0	0
$561$	−5.88844	−0.248610
$562$	0	0
$563$	−38.5652	−1.62533	−0.812665	−	0.582732i	$-0.801984\pi$
$563$	−38.5652	−1.62533	−0.812665	+	0.582732i	$0.801984\pi$
$564$	0	0
$565$	38.4113	1.61598
$566$	0	0
$567$	20.5204	0.861775
$568$	0	0
$569$	−7.83080	−0.328284	−0.164142	−	0.986437i	$-0.552486\pi$
$569$	−7.83080	−0.328284	−0.164142	+	0.986437i	$0.552486\pi$
$570$	0	0
$571$	−5.40294	−0.226106	−0.113053	−	0.993589i	$-0.536063\pi$
$571$	−5.40294	−0.226106	−0.113053	+	0.993589i	$0.536063\pi$
$572$	0	0
$573$	−14.2466	−0.595160
$574$	0	0
$575$	−28.2586	−1.17847
$576$	0	0
$577$	37.9284	1.57898	0.789489	−	0.613764i	$-0.210345\pi$
$577$	37.9284	1.57898	0.789489	+	0.613764i	$0.210345\pi$
$578$	0	0
$579$	25.3975	1.05548
$580$	0	0
$581$	−1.70681	−0.0708103
$582$	0	0
$583$	17.5111	0.725234
$584$	0	0
$585$	124.644	5.15340
$586$	0	0
$587$	30.2997	1.25060	0.625302	−	0.780383i	$-0.284976\pi$
$587$	30.2997	1.25060	0.625302	+	0.780383i	$0.284976\pi$
$588$	0	0
$589$	−2.28579	−0.0941842
$590$	0	0
$591$	39.0036	1.60439
$592$	0	0
$593$	−3.82803	−0.157198	−0.0785992	−	0.996906i	$-0.525045\pi$
$593$	−3.82803	−0.157198	−0.0785992	+	0.996906i	$0.525045\pi$
$594$	0	0
$595$	−3.11205	−0.127582
$596$	0	0
$597$	−28.0813	−1.14929
$598$	0	0
$599$	23.6279	0.965409	0.482704	−	0.875783i	$-0.339655\pi$
$599$	23.6279	0.965409	0.482704	+	0.875783i	$0.339655\pi$
$600$	0	0
$601$	5.61162	0.228903	0.114451	−	0.993429i	$-0.463489\pi$
$601$	5.61162	0.228903	0.114451	+	0.993429i	$0.463489\pi$
$602$	0	0
$603$	−27.7996	−1.13209
$604$	0	0
$605$	23.5872	0.958957
$606$	0	0
$607$	−31.5340	−1.27992	−0.639962	−	0.768406i	$-0.721050\pi$
$607$	−31.5340	−1.27992	−0.639962	+	0.768406i	$0.721050\pi$
$608$	0	0
$609$	−23.4806	−0.951481
$610$	0	0
$611$	−50.5741	−2.04601
$612$	0	0
$613$	31.4283	1.26938	0.634689	−	0.772767i	$-0.281128\pi$
$613$	31.4283	1.26938	0.634689	+	0.772767i	$0.281128\pi$
$614$	0	0
$615$	1.74716	0.0704524
$616$	0	0
$617$	−14.2555	−0.573905	−0.286952	−	0.957945i	$-0.592642\pi$
$617$	−14.2555	−0.573905	−0.286952	+	0.957945i	$0.592642\pi$
$618$	0	0
$619$	−6.08036	−0.244390	−0.122195	−	0.992506i	$-0.538993\pi$
$619$	−6.08036	−0.244390	−0.122195	+	0.992506i	$0.538993\pi$
$620$	0	0
$621$	79.4394	3.18779
$622$	0	0
$623$	−7.48581	−0.299913
$624$	0	0
$625$	−26.4765	−1.05906
$626$	0	0
$627$	26.5357	1.05973
$628$	0	0
$629$	10.3544	0.412859
$630$	0	0
$631$	5.88983	0.234470	0.117235	−	0.993104i	$-0.462597\pi$
$631$	5.88983	0.234470	0.117235	+	0.993104i	$0.462597\pi$
$632$	0	0
$633$	−10.7326	−0.426583
$634$	0	0
$635$	41.6384	1.65237
$636$	0	0
$637$	−5.61227	−0.222366
$638$	0	0
$639$	−70.4809	−2.78818
$640$	0	0
$641$	6.12352	0.241865	0.120932	−	0.992661i	$-0.461412\pi$
$641$	6.12352	0.241865	0.120932	+	0.992661i	$0.461412\pi$
$642$	0	0
$643$	37.6734	1.48569	0.742847	−	0.669462i	$-0.233475\pi$
$643$	37.6734	1.48569	0.742847	+	0.669462i	$0.233475\pi$
$644$	0	0
$645$	−25.5528	−1.00614
$646$	0	0
$647$	−40.5122	−1.59270	−0.796350	−	0.604836i	$-0.793239\pi$
$647$	−40.5122	−1.59270	−0.796350	+	0.604836i	$0.793239\pi$
$648$	0	0
$649$	−6.22754	−0.244452
$650$	0	0
$651$	−1.61492	−0.0632936
$652$	0	0
$653$	11.2530	0.440365	0.220183	−	0.975459i	$-0.429335\pi$
$653$	11.2530	0.440365	0.220183	+	0.975459i	$0.429335\pi$
$654$	0	0
$655$	−66.5568	−2.60059
$656$	0	0
$657$	−10.7602	−0.419797
$658$	0	0
$659$	−25.0186	−0.974587	−0.487294	−	0.873238i	$-0.662016\pi$
$659$	−25.0186	−0.974587	−0.487294	+	0.873238i	$0.662016\pi$
$660$	0	0
$661$	35.1914	1.36879	0.684393	−	0.729113i	$-0.260067\pi$
$661$	35.1914	1.36879	0.684393	+	0.729113i	$0.260067\pi$
$662$	0	0
$663$	17.8683	0.693947
$664$	0	0
$665$	14.0241	0.543833
$666$	0	0
$667$	−44.4857	−1.72249
$668$	0	0
$669$	−62.3652	−2.41118
$670$	0	0
$671$	17.2050	0.664190
$672$	0	0
$673$	−19.5402	−0.753221	−0.376610	−	0.926372i	$-0.622910\pi$
$673$	−19.5402	−0.753221	−0.376610	+	0.926372i	$0.622910\pi$
$674$	0	0
$675$	−61.6986	−2.37478
$676$	0	0
$677$	−29.7159	−1.14208	−0.571038	−	0.820923i	$-0.693459\pi$
$677$	−29.7159	−1.14208	−0.571038	+	0.820923i	$0.693459\pi$
$678$	0	0
$679$	13.9046	0.533611
$680$	0	0
$681$	−21.9565	−0.841375
$682$	0	0
$683$	−26.0686	−0.997486	−0.498743	−	0.866750i	$-0.666205\pi$
$683$	−26.0686	−0.997486	−0.498743	+	0.866750i	$0.666205\pi$
$684$	0	0
$685$	56.8637	2.17265
$686$	0	0
$687$	−41.8429	−1.59640
$688$	0	0
$689$	−53.1367	−2.02435
$690$	0	0
$691$	39.2377	1.49267	0.746335	−	0.665570i	$-0.231812\pi$
$691$	39.2377	1.49267	0.746335	+	0.665570i	$0.231812\pi$
$692$	0	0
$693$	13.1990	0.501390
$694$	0	0
$695$	−21.2408	−0.805709
$696$	0	0
$697$	0.176337	0.00667922
$698$	0	0
$699$	−60.3163	−2.28137
$700$	0	0
$701$	2.30117	0.0869141	0.0434570	−	0.999055i	$-0.486163\pi$
$701$	2.30117	0.0869141	0.0434570	+	0.999055i	$0.486163\pi$
$702$	0	0
$703$	−46.6613	−1.75986
$704$	0	0
$705$	89.2853	3.36268
$706$	0	0
$707$	12.4295	0.467460
$708$	0	0
$709$	−2.18085	−0.0819035	−0.0409518	−	0.999161i	$-0.513039\pi$
$709$	−2.18085	−0.0819035	−0.0409518	+	0.999161i	$0.513039\pi$
$710$	0	0
$711$	84.5920	3.17245
$712$	0	0
$713$	−3.05958	−0.114582
$714$	0	0
$715$	32.3029	1.20806
$716$	0	0
$717$	−23.1168	−0.863313
$718$	0	0
$719$	9.06273	0.337983	0.168991	−	0.985618i	$-0.445949\pi$
$719$	9.06273	0.337983	0.168991	+	0.985618i	$0.445949\pi$
$720$	0	0
$721$	15.3495	0.571647
$722$	0	0
$723$	20.3150	0.755524
$724$	0	0
$725$	34.5509	1.28319
$726$	0	0
$727$	−47.6743	−1.76814	−0.884070	−	0.467354i	$-0.845207\pi$
$727$	−47.6743	−1.76814	−0.884070	+	0.467354i	$0.845207\pi$
$728$	0	0
$729$	20.6543	0.764974
$730$	0	0
$731$	−2.57897	−0.0953868
$732$	0	0
$733$	−17.1971	−0.635189	−0.317594	−	0.948227i	$-0.602875\pi$
$733$	−17.1971	−0.635189	−0.317594	+	0.948227i	$0.602875\pi$
$734$	0	0
$735$	9.90811	0.365466
$736$	0	0
$737$	−7.20458	−0.265384
$738$	0	0
$739$	−5.99838	−0.220654	−0.110327	−	0.993895i	$-0.535190\pi$
$739$	−5.99838	−0.220654	−0.110327	+	0.993895i	$0.535190\pi$
$740$	0	0
$741$	−80.5217	−2.95804
$742$	0	0
$743$	2.05061	0.0752296	0.0376148	−	0.999292i	$-0.488024\pi$
$743$	2.05061	0.0752296	0.0376148	+	0.999292i	$0.488024\pi$
$744$	0	0
$745$	4.09421	0.150000
$746$	0	0
$747$	−12.1807	−0.445668
$748$	0	0
$749$	−14.4365	−0.527499
$750$	0	0
$751$	−33.7674	−1.23219	−0.616096	−	0.787671i	$-0.711287\pi$
$751$	−33.7674	−1.23219	−0.616096	+	0.787671i	$0.711287\pi$
$752$	0	0
$753$	83.3014	3.03567
$754$	0	0
$755$	45.7300	1.66429
$756$	0	0
$757$	13.9018	0.505269	0.252635	−	0.967562i	$-0.418703\pi$
$757$	13.9018	0.505269	0.252635	+	0.967562i	$0.418703\pi$
$758$	0	0
$759$	35.5187	1.28925
$760$	0	0
$761$	26.9919	0.978456	0.489228	−	0.872156i	$-0.337279\pi$
$761$	26.9919	0.978456	0.489228	+	0.872156i	$0.337279\pi$
$762$	0	0
$763$	3.25636	0.117888
$764$	0	0
$765$	−22.2092	−0.802975
$766$	0	0
$767$	18.8973	0.682341
$768$	0	0
$769$	−41.3033	−1.48944	−0.744718	−	0.667379i	$-0.767416\pi$
$769$	−41.3033	−1.48944	−0.744718	+	0.667379i	$0.767416\pi$
$770$	0	0
$771$	−32.3695	−1.16576
$772$	0	0
$773$	18.0477	0.649130	0.324565	−	0.945863i	$-0.394782\pi$
$773$	18.0477	0.649130	0.324565	+	0.945863i	$0.394782\pi$
$774$	0	0
$775$	2.37630	0.0853592
$776$	0	0
$777$	−32.9664	−1.18266
$778$	0	0
$779$	−0.794643	−0.0284710
$780$	0	0
$781$	−18.2659	−0.653605
$782$	0	0
$783$	−97.1279	−3.47107
$784$	0	0
$785$	76.8946	2.74449
$786$	0	0
$787$	2.45217	0.0874105	0.0437053	−	0.999044i	$-0.486084\pi$
$787$	2.45217	0.0874105	0.0437053	+	0.999044i	$0.486084\pi$
$788$	0	0
$789$	77.6928	2.76594
$790$	0	0
$791$	−12.3428	−0.438858
$792$	0	0
$793$	−52.2079	−1.85396
$794$	0	0
$795$	93.8095	3.32708
$796$	0	0
$797$	−33.8480	−1.19896	−0.599478	−	0.800391i	$-0.704625\pi$
$797$	−33.8480	−1.19896	−0.599478	+	0.800391i	$0.704625\pi$
$798$	0	0
$799$	9.01133	0.318798
$800$	0	0
$801$	−53.4226	−1.88760
$802$	0	0
$803$	−2.78863	−0.0984087
$804$	0	0
$805$	18.7717	0.661614
$806$	0	0
$807$	36.4112	1.28174
$808$	0	0
$809$	23.5930	0.829484	0.414742	−	0.909939i	$-0.363872\pi$
$809$	23.5930	0.829484	0.414742	+	0.909939i	$0.363872\pi$
$810$	0	0
$811$	−3.55235	−0.124740	−0.0623700	−	0.998053i	$-0.519866\pi$
$811$	−3.55235	−0.124740	−0.0623700	+	0.998053i	$0.519866\pi$
$812$	0	0
$813$	−13.1157	−0.459986
$814$	0	0
$815$	25.9655	0.909533
$816$	0	0
$817$	11.6219	0.406598
$818$	0	0
$819$	−40.0521	−1.39953
$820$	0	0
$821$	−37.4720	−1.30778	−0.653891	−	0.756589i	$-0.726865\pi$
$821$	−37.4720	−1.30778	−0.653891	+	0.756589i	$0.726865\pi$
$822$	0	0
$823$	43.3752	1.51196	0.755982	−	0.654592i	$-0.227160\pi$
$823$	43.3752	1.51196	0.755982	+	0.654592i	$0.227160\pi$
$824$	0	0
$825$	−27.5864	−0.960437
$826$	0	0
$827$	−13.5519	−0.471246	−0.235623	−	0.971844i	$-0.575713\pi$
$827$	−13.5519	−0.471246	−0.235623	+	0.971844i	$0.575713\pi$
$828$	0	0
$829$	−18.0498	−0.626895	−0.313447	−	0.949606i	$-0.601484\pi$
$829$	−18.0498	−0.626895	−0.313447	+	0.949606i	$0.601484\pi$
$830$	0	0
$831$	55.7586	1.93424
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	55.0793	1.90610
$836$	0	0
$837$	−6.68014	−0.230900
$838$	0	0
$839$	−56.6173	−1.95465	−0.977323	−	0.211754i	$-0.932083\pi$
$839$	−56.6173	−1.95465	−0.977323	+	0.211754i	$0.932083\pi$
$840$	0	0
$841$	25.3911	0.875557
$842$	0	0
$843$	64.6978	2.22831
$844$	0	0
$845$	−57.5654	−1.98031
$846$	0	0
$847$	−7.57932	−0.260429
$848$	0	0
$849$	9.24353	0.317237
$850$	0	0
$851$	−62.4572	−2.14101
$852$	0	0
$853$	−11.9233	−0.408245	−0.204122	−	0.978945i	$-0.565434\pi$
$853$	−11.9233	−0.408245	−0.204122	+	0.978945i	$0.565434\pi$
$854$	0	0
$855$	100.084	3.42278
$856$	0	0
$857$	4.17512	0.142619	0.0713097	−	0.997454i	$-0.477282\pi$
$857$	4.17512	0.142619	0.0713097	+	0.997454i	$0.477282\pi$
$858$	0	0
$859$	12.8623	0.438855	0.219428	−	0.975629i	$-0.429581\pi$
$859$	12.8623	0.438855	0.219428	+	0.975629i	$0.429581\pi$
$860$	0	0
$861$	−0.561419	−0.0191331
$862$	0	0
$863$	53.6300	1.82559	0.912794	−	0.408420i	$-0.133920\pi$
$863$	53.6300	1.82559	0.912794	+	0.408420i	$0.133920\pi$
$864$	0	0
$865$	−27.0266	−0.918932
$866$	0	0
$867$	−3.18379	−0.108127
$868$	0	0
$869$	21.9229	0.743684
$870$	0	0
$871$	21.8621	0.740768
$872$	0	0
$873$	99.2307	3.35845
$874$	0	0
$875$	0.980779	0.0331564
$876$	0	0
$877$	−17.8774	−0.603677	−0.301838	−	0.953359i	$-0.597600\pi$
$877$	−17.8774	−0.603677	−0.301838	+	0.953359i	$0.597600\pi$
$878$	0	0
$879$	71.5720	2.41406
$880$	0	0
$881$	−36.4616	−1.22842	−0.614211	−	0.789142i	$-0.710526\pi$
$881$	−36.4616	−1.22842	−0.614211	+	0.789142i	$0.710526\pi$
$882$	0	0
$883$	−22.3975	−0.753737	−0.376869	−	0.926267i	$-0.622999\pi$
$883$	−22.3975	−0.753737	−0.376869	+	0.926267i	$0.622999\pi$
$884$	0	0
$885$	−33.3619	−1.12145
$886$	0	0
$887$	48.6911	1.63489	0.817443	−	0.576010i	$-0.195391\pi$
$887$	48.6911	1.63489	0.817443	+	0.576010i	$0.195391\pi$
$888$	0	0
$889$	−13.3797	−0.448742
$890$	0	0
$891$	37.9526	1.27146
$892$	0	0
$893$	−40.6087	−1.35892
$894$	0	0
$895$	41.2819	1.37990
$896$	0	0
$897$	−107.780	−3.59868
$898$	0	0
$899$	3.74085	0.124764
$900$	0	0
$901$	9.46796	0.315423
$902$	0	0
$903$	8.21091	0.273242
$904$	0	0
$905$	−13.9448	−0.463541
$906$	0	0
$907$	34.5972	1.14878	0.574391	−	0.818581i	$-0.305239\pi$
$907$	34.5972	1.14878	0.574391	+	0.818581i	$0.305239\pi$
$908$	0	0
$909$	88.7035	2.94211
$910$	0	0
$911$	0.0140640	0.000465962 0	0.000232981	−	1.00000i	$-0.499926\pi$
$911$	0.0140640	0.000465962 0	0.000232981		1.00000i	$0.499926\pi$
$912$	0	0
$913$	−3.15675	−0.104473
$914$	0	0
$915$	92.1698	3.04704
$916$	0	0
$917$	21.3868	0.706255
$918$	0	0
$919$	43.9804	1.45078	0.725389	−	0.688339i	$-0.241660\pi$
$919$	43.9804	1.45078	0.725389	+	0.688339i	$0.241660\pi$
$920$	0	0
$921$	−12.9277	−0.425981
$922$	0	0
$923$	55.4273	1.82441
$924$	0	0
$925$	48.5089	1.59496
$926$	0	0
$927$	109.542	3.59784
$928$	0	0
$929$	−28.5407	−0.936389	−0.468194	−	0.883625i	$-0.655095\pi$
$929$	−28.5407	−0.936389	−0.468194	+	0.883625i	$0.655095\pi$
$930$	0	0
$931$	−4.50640	−0.147691
$932$	0	0
$933$	79.6115	2.60636
$934$	0	0
$935$	−5.75576	−0.188233
$936$	0	0
$937$	19.5088	0.637323	0.318662	−	0.947869i	$-0.396767\pi$
$937$	19.5088	0.637323	0.318662	+	0.947869i	$0.396767\pi$
$938$	0	0
$939$	59.7798	1.95084
$940$	0	0
$941$	−14.5421	−0.474059	−0.237029	−	0.971502i	$-0.576174\pi$
$941$	−14.5421	−0.474059	−0.237029	+	0.971502i	$0.576174\pi$
$942$	0	0
$943$	−1.06365	−0.0346372
$944$	0	0
$945$	40.9851	1.33325
$946$	0	0
$947$	18.5613	0.603162	0.301581	−	0.953441i	$-0.402486\pi$
$947$	18.5613	0.603162	0.301581	+	0.953441i	$0.402486\pi$
$948$	0	0
$949$	8.46201	0.274689
$950$	0	0
$951$	−75.1573	−2.43714
$952$	0	0
$953$	45.7471	1.48189	0.740947	−	0.671564i	$-0.234377\pi$
$953$	45.7471	1.48189	0.740947	+	0.671564i	$0.234377\pi$
$954$	0	0
$955$	−13.9256	−0.450621
$956$	0	0
$957$	−43.4275	−1.40381
$958$	0	0
$959$	−18.2721	−0.590038
$960$	0	0
$961$	−30.7427	−0.991701
$962$	0	0
$963$	−103.027	−3.31998
$964$	0	0
$965$	24.8252	0.799150
$966$	0	0
$967$	−2.41688	−0.0777216	−0.0388608	−	0.999245i	$-0.512373\pi$
$967$	−2.41688	−0.0777216	−0.0388608	+	0.999245i	$0.512373\pi$
$968$	0	0
$969$	14.3474	0.460906
$970$	0	0
$971$	11.1280	0.357115	0.178558	−	0.983929i	$-0.442857\pi$
$971$	11.1280	0.357115	0.178558	+	0.983929i	$0.442857\pi$
$972$	0	0
$973$	6.82534	0.218810
$974$	0	0
$975$	83.7102	2.68087
$976$	0	0
$977$	57.6117	1.84316	0.921580	−	0.388188i	$-0.126899\pi$
$977$	57.6117	1.84316	0.921580	+	0.388188i	$0.126899\pi$
$978$	0	0
$979$	−13.8451	−0.442490
$980$	0	0
$981$	23.2391	0.741968
$982$	0	0
$983$	−1.07381	−0.0342491	−0.0171245	−	0.999853i	$-0.505451\pi$
$983$	−1.07381	−0.0342491	−0.0171245	+	0.999853i	$0.505451\pi$
$984$	0	0
$985$	38.1247	1.21475
$986$	0	0
$987$	−28.6902	−0.913219
$988$	0	0
$989$	15.5562	0.494658
$990$	0	0
$991$	−24.8448	−0.789222	−0.394611	−	0.918848i	$-0.629121\pi$
$991$	−24.8448	−0.789222	−0.394611	+	0.918848i	$0.629121\pi$
$992$	0	0
$993$	28.6794	0.910114
$994$	0	0
$995$	−27.4485	−0.870177
$996$	0	0
$997$	−14.1344	−0.447641	−0.223821	−	0.974630i	$-0.571853\pi$
$997$	−14.1344	−0.447641	−0.223821	+	0.974630i	$0.571853\pi$
$998$	0	0
$999$	−136.366	−4.31443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.g.1.1	✓	6
4.3	odd	2		3808.2.a.o.1.6	yes	6
8.3	odd	2		7616.2.a.bv.1.1		6
8.5	even	2		7616.2.a.cd.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.g.1.1	✓	6	1.1	even	1	trivial
3808.2.a.o.1.6	yes	6	4.3	odd	2
7616.2.a.bv.1.1		6	8.3	odd	2
7616.2.a.cd.1.6		6	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q-3.18379 q^{3} -3.11205 q^{5} +1.00000 q^{7} +7.13652 q^{9} +1.84951 q^{11} -5.61227 q^{13} +9.90811 q^{15} +1.00000 q^{17} -4.50640 q^{19} -3.18379 q^{21} -6.03193 q^{23} +4.68484 q^{25} -13.1698 q^{27} +7.37503 q^{29} +0.507231 q^{31} -5.88844 q^{33} -3.11205 q^{35} +10.3544 q^{37} +17.8683 q^{39} +0.176337 q^{41} -2.57897 q^{43} -22.2092 q^{45} +9.01133 q^{47} +1.00000 q^{49} -3.18379 q^{51} +9.46796 q^{53} -5.75576 q^{55} +14.3474 q^{57} -3.36714 q^{59} +9.30246 q^{61} +7.13652 q^{63} +17.4657 q^{65} -3.89541 q^{67} +19.2044 q^{69} -9.87609 q^{71} -1.50777 q^{73} -14.9156 q^{75} +1.84951 q^{77} +11.8534 q^{79} +20.5204 q^{81} -1.70681 q^{83} -3.11205 q^{85} -23.4806 q^{87} -7.48581 q^{89} -5.61227 q^{91} -1.61492 q^{93} +14.0241 q^{95} +13.9046 q^{97} +13.1990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{21} - 6 q^{23} + 12 q^{25} - 10 q^{27} + 4 q^{29} + 8 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} + 18 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^13 + 6 * q^17 - 18 * q^19 - 4 * q^21 - 6 * q^23 + 12 * q^25 - 10 * q^27 + 4 * q^29 + 8 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 + 18 * q^39 - 4 * q^41 - 20 * q^43 - 24 * q^45 - 6 * q^47 + 6 * q^49 - 4 * q^51 + 14 * q^53 - 4 * q^55 - 4 * q^57 - 20 * q^59 + 2 * q^61 + 8 * q^63 + 6 * q^65 - 28 * q^67 - 4 * q^69 - 18 * q^71 + 4 * q^73 - 20 * q^75 - 8 * q^77 + 2 * q^79 + 34 * q^81 - 20 * q^83 - 4 * q^85 - 22 * q^87 - 6 * q^89 - 4 * q^91 + 40 * q^93 + 6 * q^95 + 30 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more